More intro

biblio.bib

@@ -1374,6 +1374,19 @@
 	bibsource = {dblp computer science bibliography, https://dblp.org}
 }
 
+@book{KearnsV94,
+	author    = {Michael J. Kearns and
+	             Umesh V. Vazirani},
+	title     = {An Introduction to Computational Learning Theory},
+	publisher = {{MIT} Press},
+	year      = {1994},
+	url       = {https://mitpress.mit.edu/books/introduction-computational-learning-theory},
+	isbn      = {978-0-262-11193-5},
+	timestamp = {Wed, 10 May 2017 15:11:16 +0200},
+	biburl    = {https://dblp.org/rec/bib/books/daglib/0041035},
+	bibsource = {dblp computer science bibliography, https://dblp.org}
+}
+
 @inproceedings{DBLP:journals/corr/KlinS16,
 	author    = {Bartek Klin and
 	             Michal Szynwelski},

content/introduction.tex

@@ -8,6 +8,68 @@
 \startsection
   [title={Learning and Testing}]
 
+???
+
+First off, let me sketch the problem.
+There is some fixed, but unkown to us, language $\lang \subseteq \Sigma^{*}$.
+This can be a language of natural speech, a property in model checking, a set of traces from a protocol, etc.
+We wish to infer a description of $\lang$ by only using finitely many words.
+Note that this is not an easy task, as $\lang$ is often infinite.
+\todo{Waarom sequentiele data?}
+
+Such a learning problem can be stated and solved in a variety of ways.
+In the applications we do in our research group, we often try to infer a model of a software component.
+(\in{Chapter}[chap:applying-automata-learning] describes such an application.)
+In such cases, a learning algorithm can interact with the software.
+So it makes sense to study a learning paradigm which allows for \emph{queries}, and not just a data set of samples.
+\footnote{Instead of query learning, people also use the term \emph{active learning}.}
+
+A typical query learning framework was established by \citet[DBLP:journals/iandc/Angluin87].
+In her framework, the learning algorithm may pose two types of queries to a \emph{teacher} (or \emph{oracle}):
+
+\description{Membership queries (MQs)}
+The learner poses such a query by providing a word $w \in \Sigma^{*}$ to the teacher.
+The teacher will then reply whether $w \in \lang$ or not.
+This type of query is often generalised to more general output, so the teacher replies with $\lang(w)$.
+In some papers, such a query is then called an \emph{output query}.
+
+\description{Equivalence queries (EQs)}
+The learner can provide a hypothesised description of $\lang$.
+If the hypothesis is correct, the teacher replies with \kw{yes}.
+If, however, the hypothesis is incorrect, the teacher replies with \kw{no}, together with a counterexample (i.e., a word which is in $\lang$ but not in the hypothesis or vice versa).
+
+With these queries, the learner algorithm is supposed to converge to a correct model.
+This type of learning is hence called \emph{exact learning}.
+\citet[DBLP:journals/iandc/Angluin87] showed that one can do this efficiently for deterministic finite automata DFAs (when $\lang$ is in the class of regular languages).
+
+Another paradigm which is relevant for our type of applications is PAC-learning with membership queries.
+Here, the algorithm can again use MQs as before, but the EQs are replace by random sampling (by a fixed probability distribution).
+So the allowed query is:
+
+\description{Random sample query (EX)}
+If the learner poses this query (there are no parameters), the teacher will respond with a random word $w$, together with its label $w \in \lang$.
+
+Instead of requiring that the learner exactly learns the model, we only require the following.
+The learner should \emph{probably} return a model which is \emph{approximate} to the target.
+This gives the name probably approximately correct (PAC).
+Note that there are two uncertainties: the probable and the approximate part.
+Both part are bounded by parameters, so one can determine the confidence.
+
+\startremark
+When using only EQs, only MQs, or only EXs, then there are hardness results for learning DFAs.
+See the book of \citet[KearnsV94] for such results.
+\stopremark
+
+DFAs can be efficiently learned in the PAC model.
+In fact, any exact learning algorithm with MQs and EQs can be transformed into a PAC algorithm with MQs (see exercise X in \citenp[KearnsV94]).
+For this reason, we mostly focus on the former type of learning in this thesis.
+The transformation from exact learning to PAC learning is implemented by simply \emph{testing} the hypothesis with random samples.
+The number of samples depend on the required precision (i.e., both parameters in the PAC model).
+The type of testing we will see in later chapter is slightly different, however.
+We will often choose the probability distribution ourself and even change it between hypotheses.
+Although the PAC model does not allow this and we do not acquire the promised bounds, we are able to give other guarantees with those testing methods.
+
+
 \todo{Gekopieerd van test methods paper.}
 
 Finite state machine conformance testing is a core topic in testing literature that is relevant for communication protocols and other reactive systems.

environment.tex

@@ -20,12 +20,13 @@
 \setupwhitespace[medium]
 
 \defineenumeration[definition][text=Definition]
+\defineenumeration[remark][text=Remark]
 \defineenumeration[example][text=Example]
 \defineenumeration[lemma][text=Lemma]
 \defineenumeration[proposition][text=Proposition]
 \defineenumeration[theorem][text=Theorem]
 \defineenumeration[corollary][text=Corollary]
-\setupenumeration[definition,example,lemma,proposition,theorem,corollary][alternative=serried,width=fit,right=.]
+\setupenumeration[definition,remark,example,lemma,proposition,theorem,corollary][alternative=serried,width=fit,right=.]
 
 \define\QED{\hfill$\square$}
 \definestartstop[proof][before={{\it Proof. }}, after={\QED}]